论文标题
同时谎言代数的Whittaker类别
A Whittaker category for the Symplectic Lie algebra
论文作者
论文摘要
对于任何$ n \ in \ mathbb {z} _ {\ geq 2} $，令$ \ mathfrak {m} _n $是$ \ mathfrak {sp} _ {sp} _ {2n} $ spand $ tyly long pange natund vectors $ x _ _ _ {-2n} $ x {-2m_i _i} $，一个$ \ mathfrak {sp} _ {2n} $ - 模块$ m $相对于Whittaker Pair $（\ Mathfrak {sp} _ {2n}，\ Mathfrak {m Mathfrak {m} _n）$，如果$ \ m mathfrak的动作，则根据$ \ mathfrak $ \ rapra $ {m} $ n $ ntifity fin， Mazorchuk。这种模块比Kostant定义的经典Whittaker模块更一般。在本文中，我们表明，每个非单个块$ \ MATHCAL {WH} _ {\ MATHBF {a}}^μ$具有有限的尺寸whittaker vector子空间等同于模块类别$ \ Mathcal $ \ Mathcal {w} $ \ Mathcal {d} _n^{ev} $，是半简单的。作为推论，块中的任何简单模块{ $ \ mathfrak {sp} _ {2n} $。我们还从$ u（\ mathfrak {sp} _ {2n}）$表征了所有可能的代数同态，以在自然条件下到Weyl代数$ \ Mathcal {D} _n $。
For any $n\in \mathbb{Z}_{\geq 2}$, let $\mathfrak{m}_n$ be the subalgebra of $\mathfrak{sp}_{2n}$ spanned by all long negative root vectors $X_{-2ε_i}$, $i=1,\dots,n$. An $\mathfrak{sp}_{2n}$-module $M$ is called a Whittaker module with respect to the Whittaker pair $(\mathfrak{sp}_{2n},\mathfrak{m}_n)$ if the action of $\mathfrak{m}_n$ on $M$ is locally finite, according to a definition of Batra and Mazorchuk. This kind of modules are more general than the classical Whittaker modules defined by Kostant. In this paper, we show that each non-singular block $\mathcal{WH}_{\mathbf{a}}^μ$ with finite dimensional Whittaker vector subspaces is equivalent to a module category $\mathcal{W}^{\mathbf{a}}$ of the even Weyl algebra $\mathcal{D}_n^{ev}$ which is semi-simple. As a corollary, any simple module in the block $\mathcal{WH}_{\mathbf{i}}^{-\frac{1}{2}ω_n}$ for the fundamental weight $ω_n$ is equivalent to the Nilsson's module $N_{\mathbf{i}}$ up to an automorphism of $\mathfrak{sp}_{2n}$. We also characterize all possible algebra homomorphisms from $U(\mathfrak{sp}_{2n})$ to the Weyl algebra $\mathcal{D}_n$ under a natural condition.
